Electronic transport through single molecules Matthias H. Hettler Collaborators: M. R. Wegewijs, H. Schoeller: RWTH Aachen W. Wenzel, A. Thielmann, P. Stampfuss: INT J. Heurich, J.-C. Cuevas, G. Schön:Uni Karlsruhe Forschungszentrum Karlsruhe Institut für Nanotechnologie (INT)
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Electronic transport through
single molecules
Matthias H. Hettler
Collaborators:
M. R. Wegewijs, H. Schoeller: RWTH Aachen
W. Wenzel, A. Thielmann, P. Stampfuss: INT
J. Heurich, J.-C. Cuevas, G. Schön:Uni Karlsruhe
Forschungszentrum KarlsruheInstitut für Nanotechnologie (INT)
�Molecular Orbitals severely broadened on order of coupling energy
�Electrons stay short time on molecule
�Only static e-e interactions important
�
single particle description suffices
�
If � charging energy
�
Transport via coherent scattering states
Landauer Approach: I V � 2e
�
h
�
dE T E ,V f E � �
L
� f E � �
R
F(E): Fermi function T(E,V): Transmission function, eV �
L
R
Strong coupling methodology
“extended molecule”
�Density Functional Theory (DFT) treatment of “extended molecule”
�Couple the resulting Kohn-Sham eigenstates to bulk electrodes
�“Diagonalize”, compute electron density on molecule, and iterate
Figures from Heurich,Cuevas, Wenzel and Schön, PRL 88 (2002)
Many groups, many variations in many aspects of theory
�Molecular Orbitals stay sharp
�Electrons stay long time on molecule
�Dynamic e-e interactions important
�
many-body description necessary
�If < temperature, << charging energy
�
Transport via sequential tunneling
Physical picture:Weak molecule-electrode coupling
Benzene-(1,4)-dithiolate/Au system
TheoryM. DiVentra, S.T. Pantelides, N.D. Lang, PRL 84, 979 (2000)P.S. Damle, A. W. Gosh, S. Datta, PRB 64, 201403 (2001)E. G. Emberly, G. Kirczenow, PRB 64, 235412 (2001)
2 benzenes, each one connected to different electrode!
M.A.Reed et al., Science 278 (1997): First single molecule experiment?
Bias Voltage (V)
Small current?
C C
CC
C C
SS
How to weakly coupled molecules?
Tunnel contact “Effective”Molecular Device
Au Au
C C
CC
C C
H H
HH
BufferMol.
BufferMol.
Example: J. Park, A.N. Pasupathy et al., Nature 417 (2002)
Sequential tunneling on/off molecule,transitions between many-electron states
Tunnel contact
Benzene and electron system
�
Prototype aromatic molecule
� (C,H atoms) and
� (C atoms) electron system
�
High symmetry, good energetic separation of
�
and
�
systems
� Small enough to test theoretical methods
C C
CC
C C
H
HH
H H
Sp2 ��
H Pz
� �
Electronic structure � Interacting Model
�
Effective interacting Hamiltonian for � electronsLow energy spectrum 0-5 eV well-reproduced for neutral states molecule (exp. & theory)less accurate for the charged states (anion) (not much known)
�
Wavefunction amplitudes � Tunneling
Dipole moments � Emission and Absorption of Photons
H mol. � �
ij ��
ij c i �
† c j ��
ijkl � � 'U ijkl ci �
† c j � '† ck � ' c l �
Create orthogonalized Wannier-stateslocalized at C-atom i = 1,...,6 on the ring
�
i
�
radiative relaxation of many-body states
Effective Hamiltonian
�
ij
� �
i H kin
� V nucl.
� V � eff.
� V ext�
j
H � �
ij ��
ij c i �
† c j �
ijkl � � '
U ijkl ci �
† c j � '† ck � ' c l �
U ijkl
ij e2
x � x 'kl
V � eff. il
jk � �
ije2
x � x 'kl
Freeze “core” �� electrons
� effective potentialfor remaining � electrons
e.g. partial drop of bias potential
Full unscreened Coulomb interaction for � electrons (no �� � polarization)
External Bias Potential
Adjustment of � electrons to external linear ramp potential
V ijext � d 3 r
�
i r V ext r �
j r
V ext r x
� V L
� V R
2
� V L
� V R xL
V bias
� V L� V R L � 0.4 nm
H bias
eij
V ijext ci † c j
“Worst case”: All the bias drops between the tunnel barriers
H mol. � leads
�
2 ��
e l k ��cl �† a k �� � h.c.
Electron Tunneling
�
e , a k �� : Density of states and operators in electrode α=L,R
Transition Rates
�
s ' � s
� � � �
f � E s
� E s ' � i
t i
�
s ci � s '2
�
s ' � s
� �
1 � f E s
� E s '
i
t i
s ci s '2
�
s ' � sd � 4 e2
3
� 3 c3E s
� E s '3 N E s
� E s ' s d s ' 2
d ij
� �
d 3 r
�
i r e r �
j r
N � �E
� � � 1 � N
�
E
�
:single-particle states
s,s': many-particle states, determined by effective Hamiltonian
Molecule not orthogonal to electrodeElectric field not // to transport axisMixing of y-symmetric and y-antisymmetric statesWill have at least quantitative effect
�
Low-lying anionic Rydberg statesExtra electron occupies a diffuse � orbital ?Spontaneous relaxation to lower Rydberg many-particle state ?Manipulate with symmetric side groups
�
VibrationsCould provide processes to help escape blocking state
�
Adiabatic change of nuclear lattice in blocked stateSlower than electron motion
Possible spoilers : (
� Molecular Electronics – Approach
� Strong vs. weak coupling picture
� Weakly coupled benzene - Effective � electron model